Question: In the diagram, $\triangle PQR$ is right-angled at $P$ and has $PQ=2$ and $PR=2\sqrt{3}$.  Altitude $PL$ intersects median $RM$ at $F$.  What is the length of $PF$? [asy]
draw((0,0)--(10,0)--(0,10*sqrt(3))--cycle);
draw((0,0)--(7.5,4.33)); draw((0,10*sqrt(3))--(5,0));
draw((6.68,3.86)--(7.17,3.01)--(7.99,3.49));
label("$P$",(0,0),SW); label("$M$",(5,0),S); label("$Q$",(10,0),SE);

label("$L$",(7.5,4.33),NE); label("$R$",(0,10*sqrt(3)),N); label("$F$",(4.29,2.47),NW);
[/asy]
Explanation: Since $PQ=2$ and $M$ is the midpoint of $PQ$, then $PM = MQ =\frac{1}{2}(2)=1$.

Since $\triangle PQR$ is right-angled at $P$, then by the Pythagorean Theorem,  \[ RQ = \sqrt{PQ^2+PR^2} = \sqrt{2^2+(2\sqrt{3})^2}=\sqrt{4+12}=\sqrt{16}=4. \](Note that we could say that $\triangle PQR$ is a $30^\circ$-$60^\circ$-$90^\circ$ triangle, but we do not actually need this fact.)

Since $PL$ is an altitude, then $\angle PLR = 90^\circ$, so $\triangle RLP$ is similar to $\triangle RPQ$ (these triangles have right angles at $L$ and $P$ respectively, and a common angle at $R$).

Therefore, $\frac{PL}{QP}=\frac{RP}{RQ}$ or $PL = \frac{(QP)(RP)}{RQ}= \frac{2(2\sqrt{3})}{4}=\sqrt{3}$.

Similarly, $\frac{RL}{RP} = \frac{RP}{RQ}$ so $RL = \frac{(RP)(RP)}{RQ} = \frac{(2\sqrt{3})(2\sqrt{3})}{4}=3$.

Therefore, $LQ=RQ-RL=4-3=1$ and $PF = PL - FL = \sqrt{3}-FL$.

So we need to determine the length of $FL$.

Drop a perpendicular from $M$ to $X$ on $RQ$.

[asy]
draw((5,0)--(8.75,2.17)); label("$X$",(8.75,2.17),NE);
draw((7.99,1.72)--(8.43,.94)--(9.20,1.39));
draw((0,0)--(10,0)--(0,10*sqrt(3))--cycle);
draw((0,0)--(7.5,4.33)); draw((0,10*sqrt(3))--(5,0));
draw((6.68,3.86)--(7.17,3.01)--(7.99,3.49));
label("$P$",(0,0),SW); label("$M$",(5,0),S); label("$Q$",(10,0),SE);

label("$L$",(7.5,4.33),NE); label("$R$",(0,10*sqrt(3)),N); label("$F$",(4.29,2.47),NW);
[/asy]

Then $\triangle MXQ$ is similar to $\triangle PLQ$, since these triangles are each right-angled and they share a common angle at $Q$.  Since $MQ = \frac{1}{2}PQ$, then the corresponding sides of $\triangle MXQ$ are half as long as those of $\triangle PLQ$.

Therefore, $QX=\frac{1}{2}QL=\frac{1}{2}(1)=\frac{1}{2}$ and $MX = \frac{1}{2}PL = \frac{1}{2}(\sqrt{3})=\frac{\sqrt{3}}{2}$.

Since $QX=\frac{1}{2}$, then $RX = RQ-QX = 4 - \frac{1}{2}=\frac{7}{2}$.

Now $\triangle RLF$ is similar to $\triangle RXM$ (they are each right-angled and share a common angle at $R$).

Therefore, $\frac{FL}{MX}=\frac{RL}{RX}$ so $FL = \frac{(MX)(RL)}{RX}=\frac{\frac{\sqrt{3}}{2}(3)}{\frac{7}{2}} = \frac{3\sqrt{3}}{7}$.

Thus, $PF = \sqrt{3} - \frac{3\sqrt{3}}{7} = \boxed{\frac{4\sqrt{3}}{7}}$.